Proving Trigonometric IdentitiesDate: 02/12/99 at 13:47:56 From: Hannah Calsyn Subject: Trig Identities Can you teach me how to prove trig identities? I don't know where to start or what to do exactly. Date: 02/12/99 at 15:59:29 From: Doctor Rob Subject: Re: Trig Identities Thanks for writing to Ask Dr. Math! There are usually many ways to prove trigonometric identities. Some are short and very elegant. Others are longer and more tedious, but any proof should do. The shortest proofs involve *pattern recognition*. In the equation you are trying to prove, you detect a pattern that appears in one of the standard identities you know, and that allows you to make a substitution which simplifies the identity a lot. There is a knack to this which is a bit hard to teach, but it comes with practice. See the following web page for a list of standard identities: http://mathforum.org/dr.math/faq/formulas/faq.trig.html There is a systematic way to produce a proof, which may be fairly long, but still valid. I will describe it below. A proof of an identity is often constructed by starting with one expression on the lefthand side of the equation and another on the righthand side which you want to prove as equal. You make substitutions on both sides until they are both reduced to the same identical expression. Then, since the last equation is identically true, you can reverse the steps to conclude that the first equation that you were trying to prove is also true. Alternatively, you can rearrange your work to start with the lefthand side of the first equation, work down the chain of equal expressions on the lefthand sides until you reach the bottom, then work up the chain of equal expressions on the righthand side until you are back at the top. Here is an example: cos^4(x) - sin^4(x) = cos(2*x) [cos^2(x) - sin^2(x)]*[cos^2(x) + sin^2(x)] = 2*cos^2(x) - 1 Here we factored the lefthand side as the difference of two squares, and we used the cosine double-angle formula on the righthand side. [cos^2(x) - sin^2(x)]*1 = 2*cos^2(x) - 1 Here we used the identity cos^2(x) + sin^2(x) = 1. cos^2(x) - [1 - cos^2(x)] = 2*cos^2(x) - 1 Here we used 1 - cos^2(x) = sin^2(x). 2*cos^2(x) - 1 = 2*cos^2(x) - 1 Here we just expanded and combined like terms. Now we have an obviously true equation at the bottom. To prove the original identity, we can just reverse the steps, (1.) 2*cos^2(x) - 1 = 2*cos^2(x) - 1 (2.) cos^2(x) - [1 - cos^2(x)] = 2*cos^2(x) - 1 (3.) cos^2(x) - sin^2(x) = 2*cos^2(x) - 1 (4.) [cos^2(x) - sin^2(x)]*[cos^2(x) + sin^2(x)] = 2*cos^2(x) - 1 (5.) cos^4(x) - sin^4(x) = cos(2*x) and supply the reasons. Alternatively, we can use this form: cos^4(x) - sin^4(x) = [cos^2(x)] - sin^2(x)]*[cos^2(x) + sin^2(x)] = cos^2(x) - sin^2(x) = cos^2(x) - [1 - cos^2(x)] = 2*cos^2(x) - 1 = cos(2*x) by working down the left and up the right of our original derivation, and supply the reasons. Here is a systematic way to produce a trigonometry proof: 1. Convert all cosecants, secants, cotangents, and tangents to expressions involving only sines and cosines. The first group of five identities on the above web page will allow you to do this easily. That will give you a "simpler" equation to prove. 2. Look at all the angles in the sines and cosines in the new equation you are trying to prove. If any is a sum or difference of angles, use a sine or cosine sum-of-angles formula, as in the 9th and 10th groups of identities from the Dr. Math FAQ web page. That will give you a "simpler" equation to prove. 3. Look at all the angles in the sines and cosines in the new equation you are trying to prove. If any is a multiple of some angle, use a sine or cosine multiple-angle formula, as in the 11th through 16th groups of identities from the web page. Likewise for half-angles and the 12th group of identities. That will give you a "simpler" equation to prove. 4. Expand all the expressions in sight, combine like terms, and simplify. That will give you a "simpler" equation to prove. 5. Replace all occurrences of the square or higher power of a cosine using the identity cos^2(u) = 1 - sin^2(u). Expand, combine, and simplify again. That will give you a "simpler" equation to prove. 6. Factor numerator and denominator if possible and cancel common factors if any. That will give you a "simpler" equation to prove. 7. At this point, you should have an equation that is obviously true, of the form E = E, where E is some expression involving sines to powers and possibly some cosines to the first power. 8. From this sequence of equations, construct a proof of the original equation using one of the two methods described above: either working from the bottom up, or else working from the top left down, over, and up, to the top right. I did warn you that this was tedious! It will work, however, in almost all cases, and with minor variations in essentially all cases. Be sure to do the algebra correctly when you are doing "expand, combine, simplify, and factor" operations! - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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